Given the component parameters, solve the first-order equation
$$
\dot a=aH_0E(a^{-1}-1),
\qquad
t-t_i=\int_{a_i}^a\frac{d\widetilde a}{\widetilde aH(\widetilde a)}.
$$
A <cosmic string network> has $w=-1/3$, so $\rho\propto a^{-2}$ and $H\propto a^{-1}$. Therefore $\dot a$ is constant and a string-dominated expanding universe has
$$
\boxed{a(t)\propto t}
$$
after choosing the Big Bang as $t=0$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 310 / 1 / b / Solution

{wiki=Cosmic_string}

An isotropic network of long cosmic strings has average equation-of-state parameter $w=-1/3$ and energy density proportional to $a^{-2}$.


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 Incoming links: Cosmic string network